M 1310
3.6
Combing Functions
Methods for combining functions:
1. Sum
(f + g)(x) = f (x) + g(x)
2. Difference
(f − g)(x) = f (x) − g(x)
3. Product
(fg)(x) = f (x)g(x)
4. Quotient
⎞ f (x)
⎛f
provided g( x ) ≠ 0 .
⎜ (x) ⎟ =
g
g
(
x
)
⎠
⎝
5. Composition
(f o g)(x) − f (g(x))
Example 1:
Let f ( x ) = x 2 + 5 and g( x ) = x + 5 .
a. Find f +g
b. Find f – g
c. Find f • g
d. Find f / g
e. Find f (– 1 ) – g(2)
1
M 1310
3.6
Combing Functions
2
Example 2: Suppose f ( x ) = 2 x − 3 , g( x ) = x2 − 4 x + 5 , and
h( x ) = 2 x2 .
a. (f − g)(x )
b.
(f + g)(2 )
⎛ g⎞
c. ⎜ ⎟(x )
⎝h⎠
Composition of Functions:
Machine picture:
This combined “machine” is called (f o g) (read “f composed with
g”).
The new function (f o g) is defined whenever x is in the domain of
g and g( x ) is in the domain of f.
(f o g)(x ) = f (g(x )) the second function is put into the first function.
M 1310
3.6
Combing Functions
Example 3:
Let f ( x ) = x 2 and g( x ) = x − 3
Find
a.
(f o g)(x )
b.
(g o f )(x )
c.
(f o g)(5)
Example 4:
Let f ( x ) = 2 x + 3 and g( x ) = 4 x − 1
Find
a. (f o g)(x )
b.
(f o f )(x )
c.
(g o g)(− 1)
3
M 1310
3.6
Combing Functions
Example 5:
Let f ( x ) = 2 x 2 − 3x + 1 and g( x ) = x − 4
Find
a.
(g o f )(− 5)
b.
(f o g)(− 1)
c.
(f o g)(a)
Example 6:
Let f ( x ) =
Find
a.
(f o g)(x )
b.
(g o f )(2 )
3x − 7
and g( x ) = 2 x + 1
4 + 8x
4
M 1310
3.6
Combing Functions
Example 7:
Let f ( x ) = x + 5 and g( x ) = x 2
Find
a.
(f o g)(x )
b.
(g o f )(− 1)
Example 8:
Let f ( x ) =
Find
a.
(f o g)(x )
b.
(g o f )(− 1)
1
and g( x ) = x 2 + 2
x
5